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minimal cardinality of a set such that there exist subsets

Source: Mongolian MO 2002 Grade 10 P1

April 9, 2021
combinatorics

Problem Statement

Let n,kn,k be given natural numbers. Find the smallest possible cardinality of a set AA with the following property: There exist subsets A1,A2,…,AnA_1,A_2,\ldots,A_n of AA such that the union of any kk of them is AA, but the union of any k−1k-1 of them is never AA.
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